Introduction

In this post, you'll learn how to square numbers from 1-100 in your head!

As a refresher, squaring a number simply means to multiply it by itself. For example, 4 squared is 16 because 4 times 4 is 16. You should know the squares of all the numbers from 1 through 10 by heart already.

Multiples of 10

If you already know your squares of the numbers 1 through 10, the multiples of 10 are easy. When a number from 1-100 ends in zero, simply drop the ending 0, square the remaining number, and then add 2 zeroes. For example, to work out 20 squared, drop the zero leaving the 2, square it to get 4, then tack on 2 zeroes to that 4, resulting in 400.

70 squared? 4900, because 7 squared is 49, and the two zeroes added make it 4900. 100 is trickier, but uses the same approach. 100 with the final zero dropped gives us 10. 10 squared is 100, and adding 2 zeroes gives us 10,000.

Multiples of 5

Multiples of 5 are almost as easy. You do need to make sure you know your multiplication tables up to at least 10 times 10. The method taught here is also taught in the root extraction tutorial, as well.

When given a number ending in 5, simply take the 10s digit, and multiply by a number one higher than itself. Take that answer, take a "25" on the end, and you've got the answer!

For example, let's say you're asked what 35 squared is. Take the 3 (the 10s digit), and multply it by 4 (which is one higher than 3), and you get 12. Tack a 25 on the end, giving you 1225. Simple, isn't it?

Let's try a higher number, like 75 squared. 7 times 8? 56. Tacking on the 25, gives us 5625!

To quiz yourself on squaring multiples of 10 and 5, click here. To learn the mental math approach to squaring the remaining numbers, click here. To learn the memorization approach to squaring the remaining numbers, click here.

Numbers from 1-25

For the approach using pure mental math, you'll need to know your squares from 1-25 by heart. From 1 to 10 you should already know, and from the techniques on the first page, 15, 20, and 25 will be easily handled, as well. That leaves these squares to learn by heart:

Number

Square of Number

11

121

12

144

13

169

14

196

16

256

17

289

18

324

19

361

21

441

22

484

23

529

24

576

The must be known by heart, because the method we're going to use to work out the remaining numbers requires that you can give the above numbers quickly.

Numbers from 26-50

To work out the numbers from 26 to 50, we're going to use an approach in which we multiply by 50.

Multiplying any number by 50 is easy – all you have to do is divide the number by 2, and add 2 zeroes (more accurately, you would move the decimal 2 places to the right). 48 times 50? Half of 48 is 24, and two zeroes added results in 2400, which is the correct answer. This method only involves multiplying even numbers by 50, so you won't have to worry about dealing with numbers like 24.5 (Half of 49).

When given any number from 26-50, you're first going to work out how far that number is from 50, then subtract that distance from the given number. For example, if you're given the number 47, it's easy to work out that it's only 3 away from 50. Subtracting that 3 from 47, we get 44.

Instead of solving 47 times 47, then, we're going to work out the much easier problem of 44 times 50, which is 2200. However, this isn't the same as the answer to 47 squared, so we need to make an adjustment.

From 47, we both moved up 3 to 50 and down 3 to 44. So, we square this 3 to get 9, and add that to the other answer we worked out, 2,200, to get a total of 2,209. This is the answer to 47 times 47!

So, when given a number, you work out how far the given is from 50, and find a number that's equally far below the given number (the low number), and also remember this difference. Multiply 50 times the low number, adjust it by squaring the difference you moved, and adding that amount, and the result will be the square!

Let's try this with 44, to help make this clearer. 44 is 6 away from 50, so we figure out that 44 - 6 = 38. 38 times 50 is easy, 1,900. We moved a difference of 6 in both directions, so we add 36 (6 squared) to 1900, to get 1,936!

How about 39 squared? That's 11 away from both 50 and 28. 28 times 50? 1,400. 11 squared is 121, and adding that to 1,400, we get 1,521!

Numbers from 51-75

The same technique is going to be used for numbers from 51 to 75, but with one minor change. You'll be moving down to 50, and up to another number (Previously, you moved up to 50, and down to another number). Other than that, the process is basically the same.

How about 67? The distance makes this a little more challenging, but the process is still the same. 67 is 17 away from 50 and 84, so we multiply those two numbers to get 4,200. 17 squared is 289, so we work out 4,200 plus 289 to get our final answer of 4,489.

Numbers from 76-100

As easy as multiplying by 50 has been, multiplying by 100 is even easier – just add 2 zeroes!

For numbers from 76-100, we're going to adjust upward to 100, as opposed to using 50 as we have been. Wait until you see how easy this makes the process!

Let's try working out 98 squared. 98 is 2 away from 100 and 96, and multiplied together, that gives us 9,600. 2 squared is 4, and 9600 plus 4 is 9,604. That's 98 squared!

How about 91? We start out with 8,200 (do you see why?), and add 81 (again, do you see why?), to get 8,281.

The more you practice each of these stages, the more you'll get a feel for certain patterns. This will allow you to speed up your calculations.

Numbers from 100-125

By now, you've probably figured out that you can go up to 125 with just a minor adaptation, similar to that we used when going above 50.

What's 103 squared? It's between 106 and 100, so we multiply those to get 10,600. 3 squared is 9, so that added in gives us 10,609!

What about a toughie, like 124? That's between 100 and 148, so we start with 14,800. 24 squared is 576, so we add those together to get 15,376.

With a little practice, you should have this process down in a faster time than you may have ever thought possible.

To practice squaring numbers, click here. To learn an alternative approach using memorization for squaring the numbers from 1 to 100, click here.

Prerequisites:

Links

With the exception of the multiples of 10 and 5, Jim Wilder put in some amazing work developing major system mnemonics for all the squares from 26 to 99 (Thanks again, Jim, for both your work and willingness to share it with us!):

Number

Square of Number

Number Mnemonic

Square Mnemonic

26

676

iNCH

SHaKiSH

27

729

kNocK

Key, NaP

28

784

kNiFe

CoVeR

29

841

kNoB

FoRT

31

961

MaiD

PuSHeD

32

1,024

MooN

DoSe NeaR

33

1,089

MuM

ToSS FiB

34

1,156

MaRRy

TighT LeaSH

36

1,296

MaTCH

DowN PuSH

37

1,369

MoCHa

DaMn, CHeaP

38

1,444

huMVee

TiRe RoaR

39

1,521

MoP

TaiL kNoT

41

1,681

RaT

TouCH, FiT!

42

1,764

RuN

TaKe SHaRe

43

1,849

RaM

TuFF RoPe

44

1,936

RoaR

TiP MatCH

46

2,116

ReaCH

NoT TouCH

47

2,209

RoCK

NoN SouP

48

2,304

ReeF

NaM, SiR!

49

2,401

RiB

uNRaiSeD

61

3,721

SHaDow

MaKe NighT

62

3,844

CHaiN

MoVe ReaR

63

3,969

CHuM

MoP SHiP

64

4,096

CHaiR

RiSe, PuSH

66

4,356

CHeeCH

RuM LuSH

67

4,489

CHeCK

RaRe FiB

68

4,624

CHeF

ReaCH NeaR

69

4,761

CHiP

RocK SHeeT

71

5,041

KiT

LooSe, RighT?

72

5,184

CaN

LeaD FeaR!

73

5,329

GuM

LiMe NuB

74

5,476

CaR

LuRe CaSH

76

5,776

CoaCH

LuCK, CoaCH

77

5,929

CoKe

LeaP, NaP

78

6,084

CaVe

CHooSe FiRe

79

6,241

CaP

SHiN, Right?

81

6,561

FaT

JeLLo JeT

82

6,724

FiN

CHiC NoiR

83

6,889

FoaM

CheF FiB

84

7,056

FouR

CaSe LatCH

86

7,396

FiSH

CoMb PuSH

87

7,569

FaKe

CoaL CHiP

88

7,744

FiFe

KicK ReaR

89

7,921

FiB

CaP NoD

91

8,281

PaT

FuN FighT

92

8,464

PaN

VeRy CHaR'd

93

8,649

PaM

FiSH RuB

94

8,836

PouR

ViVa MuCHo!

96

9,216

PiTCH

BaNNeD SHow

97

9,409

PuCK

PooR SOB

98

9,604

PuFF

PuSHeS aiR

99

9,801

PaPa

PuFFS iT

Memorizing the 50s

Like multiples of 10 and 5, the squares of numbers in the 50s have their own trick that is easy to remember.

Number

Square of Number

51

2,601

52

2,704

53

2,809

54

2,916

56

3,136

57

3,249

58

3,364

59

3,481

When given a number in the 50s, simply take the ones digit and add it to 25. Next, take the square of the digit in the ones place, and tack that on to the right of the previous answer.

For an example, let's use 53. The ones digit is a 3, so we add 25 to get 28. 3 squared is 9, so we add 09 to the end of the other digit to get 2,809.

57? 7 plus 25 is 32. 7 squared is 49. Therefore, 57 squared is 3,249. Once the pattern clicks, you'll find that these are quick and easy.

To practice squaring numbers, click here. To learn an approach using mathematics for squaring the numbers from 1 to 125, click here.

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8 Response to Squaring 2-Digit Numbers Mentally

I'm afraid at my age memorizing anything new seems hard, and since any two digit number is within three of a multiple of ten or five, it seems easiest to know how to square (x+/- a) where a is less than or equal to three and you would be done... 77 ^2 = (75+2)^2 = 5625 + 300 + 4

11 squared. 11 times 11. take one of the elevens and add a 0 to the end. so now it looks like 110 times 11. now take the other eleven and make the tens place equivelent to the ones place. in this case. 11 stays the same. then you separate that number. so its 11 plus 1. square the single number ( the ones place that you separated) and then add the 2 numbers back together. in this case. 10 plus 1 squared is still 11. so you take that number that you have after doing all that and add it to the first number (110) 11 plus 110 is 121.

this doesnt work for anything past 20 or anything below 11. another trick i use (im a freshman, and we do alot of squaring in my algebra honors class) is if i know what the square of the number below the number im trying to square, exapmle: if i need to square 21, and i dont have a calculator and i dont want to do it on paper, and i know 20 squared is 400, than i take 20. multiply it by 2, and add one. so 441 is 21 squared. now that i know what 21 squared is i can find 22 squared....... 441 + 21*2+1 441+43.....484

also ive been taught that if its just easier to do, lets say im squaring 88, so (90-2)^2 instead of having to memorize your formula, becuase (90-2)^2 is just 90^2 - 2(90*2)+ -2^2....... based of of (x+/- a)^2 is equivelent to x^2 +/- 2ax + a^2. I do like the 5 formula of yours, that will be helpful. lol ive been typing for like 3 hours now and i forgot how i even got here.